A corollary of our main theorem implies that a generic projective structure is not metrizable in the class of
Riemannian or pseudo-Riemannian metrics.
This result was expected: indeed, the metrization problem can be reduced to an overdetermined system of PDEs
of finite type (see e.g. M. Eastwood et al [1]}).
Nevertheless, this result was formally established only in dimension 2 (R. Bryant et al [2]})
and in dimension 3 (M. Dunajski et al [3]}); we proved it in any dimension.
